We have
\(e_2(I)+1&=& \widetilde{r}_J(I)\nonumber \\&\le & r_J(I) \hspace{28.45274pt}(\mbox{by Proposition \ref {mafi}})\nonumber \\&\le & e_1(I)-e_0(I)+\lambda (R/I)+1 \mbox{ (by }(\ref {rossi-bound}) )\) 
By Theorem REF , \(e_1(I)=e_0(I)-\lambda (R/\widetilde{I})+1\) . Substituting the value in equation (REF ), we get \(e_2(I)+1\le r_J(I)\le \lambda (\widetilde{I}/I)+2\)  which implies
\(1\le e_2(I)\le r_J(I)-1 \le \lambda (\widetilde{I}/I)+1.\) 
Moreover if \(I\)  is Ratliff-Rush closed, then we obtain \(e_2(I)=1\)  and \(r_J(I)=e_2(I)+1=2\) . Then by [1]} we have \(\operatorname{depth}G(I)\ge 1\) .
